Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation

This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\ Consider the equation $$\begin{array}{rl}{\displaystyle }& {\displaystyle y_t-\mathrm{\Delta }a(y)=mu+f\text{ on }Q}\\ {\displaystyle }& {\displaystyle y(0)=y_0,y{\mid }_{\mathrm{\Sigma }}=0}\end{array}$$ on the $n+1$-dimensional cylinder $Q=\mathrm{\Omega }\times (0,T)$ with lateral boundary $\mathrm{\Sigma }=\mathrm{\partial }\mathrm{\Omega }\times (0,T)$. The exact controllability in finite time is proved when $\mathrm{\parallel }{y}_0-y_s{\mathrm{\parallel }}_{W_0^{1,n}(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})}$ is sufficiently small, $n>1$, for every stationary solution $y_s$ such that $a(y_s)\in W^{2,q}(\mathrm{\Omega })$, where $q>n$. It is assumed that $\mathrm{\Omega }$ is a bounded open set with $C^2$ boundary and that $a\in C^2(\mathbb{R})$, $a^{\mathrm{\prime }}>0$.